CONVERGENCE BEHAVIOUR OF SOLUTIONS TO DELAY CELLULAR NEURAL NETWORKS WITH NON-PERIODIC COEFFICIENTS

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1 Electronic Journal of Differential Equations, Vol. 2007(2007), No. 46, pp ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp) CONVERGENCE BEHAVIOUR OF SOLUTIONS TO DELAY CELLULAR NEURAL NETWORKS WITH NON-PERIODIC COEFFICIENTS BING XIAO, HONG ZHANG Abstract. In this note we studied delay neural networks without periodic coefficients. Sufficient conditions are established to ensure that all solutions of the networks converge to a periodic function. An example is given to illustrate our results. 1. Introduction Let n be the number of units in a neural network, x i (t) be the state vector of the i-th unit at time t, a ij (t) be the strength of the j-th unit on the i-th unit at time t, b ij (t) be the strength of the j-th unit on the i-th unit at time t τ ij (t), and τ ij (t) 0 denote the transmission delay of the i-th unit along the axon of the j-th unit at the time t. It is well known that the delayed cellular neural networks are described by the differential equations x i(t) = c i (t)x i (t) a ij (t)f j (x j (t)) b ij (t)g j (x j (t τ ij (t))) I i (t), (1.1) for i = 1, 2,..., n, for any activation functions of signal transmission f j and g j. Here I i (t) denotes the external bias on the i-th unit at the time t, c i (t) represents the rate with which the i-th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time t. Since the cellular neural networks (CNNs) were introduced by Chua and Yang [2] in 1990, they have been successfully applied to signal and image processing, pattern recognition and optimization. Hence, CNNs have been the object of intensive analysis by numerous authors in recent years. In particular, extensive results on the problem of the existence and stability of periodic solutions for system (1.1) are given out in many literatures. We refer the reader to [3, 5, 4, 6, 7, 8, 9] and the references cited therein. Suppose that the following condition holds: (H0) c i, I i, a ij, b ij, : R R are continuous periodic functions, where i, j = 1, 2,..., n Mathematics Subject Classification. 34C25, 34K13, 34K25. Key words and phrases. Cellular neural networks; convergence; delays. c 2007 Texas State University - San Marcos. Submitted October 7, Published March 15, Supported by grants 06C586 and 06C576 from Scientific research Fund of Hunan Provincial Educational Department of China. 1

2 2 B. XIAO, H. ZHANG EJDE-2007/46 Under this assumption, most of reference above obtain that all solutions of system (1.1) converge to a periodic function. However, to the best of our knowledge, few authors have considered the convergence behavior for solutions of (1.1) without assuming (H0). Thus, it is worth while to consider the convergence behavior for solutions of (1.1) in this case. The main purpose of this paper is to give the new criteria for the convergence behavior for all solutions of (1.1). By applying mathematical analysis techniques, without assuming (H0), we derive some sufficient conditions ensuring that all solutions of system (1.1) converge to a periodic function. These results are new and complement of previously known results. An example is provided to illustrate our results. Consider the delay cellular neural networks x i(t) = c i (t)x i (t) a ij(t)f j (x j (t)) b ij(t)g j (x j (t τ ij (t))) Ii (t), (1.2) where i = 1, 2,..., n. For the rest of this paper this paper, for i, j = 1, 2,..., n, it will be assumed that c i, I i, a ij, b ij, τ ij : R R are continuous ω-periodic functions. Then, we can choose a constant τ such that τ = 1 i,j n We also use the following conditions: t [0,ω] τ ij(t) }. (1.3) (H1) For each j 1, 2,..., n}, there exist nonnegative constants L j and L j such that f j (u) f j (v) L j u v, g j (u) g j (v) L j u v, u, v R. (1.4) (H2) There exist constants η > 0, λ > 0 and ξ i > 0, i = 1, 2,..., n, such that for all t > 0 and i = 1, 2,..., n, [c i (t) λ]ξ i a ij(t) L j ξ j b ij(t) e λτ L j ξ j < η < 0, (H3) For i, j = 1, 2,..., n, c i, I i, a ij, b ij : R R are continuous functions, and lim (c i(t) c i (t)) = 0, lim (I i(t) Ii (t)) = 0, t t lim (a ij(t) a ij(t)) = 0, lim (b ij(t) b ij(t)) = 0. t t The following lemma will be useful to prove our main results in Section 2. Lemma 1.1 ([7]). Let (H1) and (H2) hold. Then system (1.2) has exactly one ω-periodic solution. As usual, we introduce the phase space C([ τ, 0]; R n ) as a Banach space of continuous mappings from [ τ, 0] to R n equipped with the supremum norm, ϕ = sup 1 i n τ t 0 ϕ i (t) for all ϕ = (ϕ 1 (t), ϕ 2 (t),..., ϕ n (t)) T C([ τ, 0]; R n ). The initial conditions associated with system (1.1) are of the form x i (s) = ϕ i (s), s [ τ, 0], i = 1, 2,..., n, (1.5) where ϕ = (ϕ 1 (t), ϕ 2 (t),..., ϕ n (t)) T C([ τ, 0]; R n ).

3 EJDE-2007/46 CONVERGENCE BEHAVIOUR OF SOLUTIONS 3 For Z(t) = (x 1 (t), x 2 (t),..., x n (t)) T, we define the norm Z(t) ξ = i=1,2,...,n ξ 1 i x i (t). The remaining part of this paper is organized as follows. In Section 2, we present some new sufficient conditions to ensure that all solutions of system (1.1) converge to a periodic function. In Section 3, we shall give some examples and remarks to illustrate our results obtained in the previous sections. 2. Main Results Theorem 2.1. Assume (H1) (H3) and that Z (t) = (x 1(t), x 2(t),..., x n(t)) T is the ω-periodic solution of (1.2). Then every solution Z(t) = (x 1 (t), x 2 (t),..., x n (t)) T of (1.1) with initial value ϕ = (ϕ 1 (t), ϕ 2 (t),..., ϕ n (t)) T C([ τ, 0]; R n ), satisfies Proof. Set lim x i(t) x i (t) = 0, i = 1, 2,..., n. t δ i (t) = [c i (t) c i (t)]x i (t) [a ij (t) a ij(t)]f j (x j (t)) [b ij (t) b ij(t)]g j (x j (t τ ij (t))) [I i (t) Ii (t)], where i = 1, 2,..., n. Since Z (t) = (x 1(t), x 2(t),..., x n(t)) T is ω-periodic, together with (H 2 ) and (H3), then for all ɛ > 0, we can choose a sufficient large constant T > 0 such that and [c i (t) λ]ξ i δ i (t) < 1 ηɛ, for all t T, (2.1) 4 a ij (t) L j ξ j b ij (t) e λτ L j ξ j < 1 η < 0, (2.2) 2 for all t T, i = 1, 2,..., n. Let Z(t) = (x 1 (t), x 2 (t),..., x n (t)) T be a solution of (1.1) with initial value ϕ = (ϕ 1 (t), ϕ 2 (t),..., ϕ n (t)) T C([ τ, 0]; R n ). Define Then for i = 1, 2,..., n, u i(t) = c i (t)u i (t) u(t) = (u 1 (t), u 2 (t),..., u n (t)) T = Z(t) Z (t). a ij (t)[f j (x j (t)) f j (x j (t))] b ij (t)[g j (x j (t τ ij (t))) g j (x j (t τ ij (t)))] δ i (t). Let i t be an index such that (2.3) ξ 1 i t u it (t) = u(t) ξ. (2.4)

4 4 B. XIAO, H. ZHANG EJDE-2007/46 Calculating the upper right derivative of e λs u is (s) along (2.3), in view of (2.1) and (H1), we have D (e λs u is (s) ) Let s=t = λe λt u it (t) e λt sign(u it (t)) c it (t)u it (t) a itj(t)[f j (x j (t)) f j (x j (t))] } b itj(t)[g j (x j (t τ itj(t))) g j (x j (t τ itj(t)))] δ it (t) e λt [c it (t) λ] u it (t) ξ 1 i t ξ it b itj(t)l j u j (t τ itj(t)) ξ 1 j ξ j } 1 4 ηɛeλt. M(t) = a itj(t) L j u j (t) ξ 1 j ξ j (2.5) τ s t eλs u(s) ξ }. (2.6) It is obvious that e λt u(t) ξ M(t), and M(t) is non-decreasing. Now, we consider two cases. Case (i). If M(t) > e λt u(t) ξ for all t T. (2.7) Then, we claim that M(t) M(T ) (2.8) which is a constant for all t T. By way of contradiction, assume that (2.8) does not hold. Consequently, there exists t 1 > T such that M(t 1 ) > M(T ). Since e λt u(t) ξ M(T ) for all τ t T. There must exist β (T, t 1 ) such that e λβ u(β) ξ = M(t 1 ) M(β), which contradicts (2.7). This contradiction implies (2.8). It follows that there exists t 2 > T such that u(t) ξ < e λt M(t) = e λt M(T ) < ɛ for all t t 2. (2.9)

5 EJDE-2007/46 CONVERGENCE BEHAVIOUR OF SOLUTIONS 5 Case (ii). If there is a point t 0 T such that M(t 0 ) = e λt0 u(t 0 ) ξ, Then, using (2.1), (2.2) and (2.5), we get D (e λs u is (s) ) s=t0 [c it0 (t 0 ) λ]e λt0 u it0 (t 0 ) ξ 1 i t0 ξ it0 a it0 j(t 0 ) L j e λt0 u j (t 0 ) ξ 1 j ξ j b it0 j(t 0 )L j e λ(t0 τi t 0 j(t 0)) u j (t 0 τ it0 j(t 0 )) ξ 1 j e λτi t 0 j(t 0) ξ j } 1 4 ηɛeλt0 [c it0 (t 0 ) λ]ξ it0 < 1 2 ηm(t 0) 1 2 ηɛeλt0. a it0 j(t 0 ) L j ξ j b it0 j(t 0 )e λτ L j ξ j }M(t 0 ) 1 4 ηɛeλt0 In addition, if M(t 0 ) ɛe λt0, then M(t) is strictly decreasing in a small neighborhood (t 0, t 0 δ 0 ). This contradicts that M(t) is non-decreasing. Hence, e λt0 u(t 0 ) ξ = M(t 0 ) < ɛe λt0, and u(t 0 ) ξ < ɛ. (2.10) Furthermore, for any t > t 0, by the same approach used in the proof of (2.10), we have e λt u(t) ξ < ɛe λt, and u(t) ξ < ɛ, if M(t) = e λt u(t) ξ. On the other hand, if M(t) > e λt u(t) ξ, t > t 0. We can choose t 0 t 3 < t such that M(t 3 ) = e λt3 u(t 3 ) ξ, u(t 3 ) ξ < ɛ, M(s) > e λs u(s) ξ for all s (t 3, t]. Using a similar argument as in the proof of Case (i), we can show that M(s) M(t 3 ) which is a constant for all s (t 3, t] which implies that u(t) ξ < e λt M(t) = e λt M(t 3 ) = u(t 3 ) ξ e λ(t t3) < ɛ. In summary, there must exist N > 0 such that u(t) ξ ɛ holds for all t > N. This completes the proof of Theorem 2.1. Remark 2.2. Without assumption (H3), we suppose that (H1*) For each j 1, 2,..., n}, there exist nonnegative constants L j and L j such that f j (u) L j u, g j (u) L j u, for all u R (H2*) There exist constants K > 0, η > 0, λ > 0 and ξ i > 0, i = 1, 2,..., n, such that for all t > K, there holds [c i (t) λ]ξ i a ij (t) L j ξ j b ij (t) e λτ L j ξ j < η < 0, are satisfied for i = 1, 2,..., n. Moreover, assume that I i (t) converge to zero as t. Then, applying the similar mathematical analysis techniques in this paper, we find that the sufficient conditions can be established to ensure the convergence to zero of all solutions of (1.1).

6 6 B. XIAO, H. ZHANG EJDE-2007/46 Remark 2.3. If the original system has asymptotically periodic coefficients, there may not be sufficient conditions ensuring the existence and the convergence of periodic solutions, which is the case in the example x (t) = x(t) 1 t 2 1 sin2 t. 3. An Example In this section, we illustrate the results obtained in previous sections. Consider the cellular neural networks with time-varying delays: x 1(t) = (1 2 1 t )x 1(t) ( 1 4 t 1 t 2 )f 1(x 1 (t)) ( t 1 t 2 )f 2(x 2 (t)) ( 1 4 t 2 t 2 )g 1(x 1 (t sin 2 t)) ( t 1 t 2 )g 2(x 2 (t 2 sin 2 t)) (cos t t 1 t 2 ), x 4 2(t) = (1 1 2 t )x 2(t) (1 t 1 t 2 )f 1(x 1 (t)) ( 1 4 5t 1 t 2 )f(x 2(t)) t (1 1 6t 2 )g 1(x 1 (t 5 sin 2 t)) ( 1 4 t 8 t 2 )g 2(x 2 (t sin 4 t)) (sin t t 1 t 6 ), (3.1) where f 1 (x) = f 2 (x) = g 1 (x) = g 2 (x) = arctan x. Noting the following cellular neural networks x 1(t) = x 1 (t) 1 4 f 1(x 1 (t)) 1 36 f 2(x 2 (t)) 1 4 g 1(x 1 (t sin 2 t)) 1 36 g 2(x 2 (t 2 sin 2 t)) cos t, x 2(t) = x 2 (t) f 1 (x 1 (t)) 1 4 f(x 2(t)) g 1 (x 1 (t 5 sin 2 t)) (3.2) 1 4 g 2(x 2 (t sin 4 t)) sin t, where c 1(t) = c 2(t) = L 1 = L 2 = L 1 = L 2 = 1, a 11(t) = b 11(t) = 1 4, a 12(t) = b 12(t) = 1 36, a 21(t) = b 21(t) = 1, a 22(t) = b 22(t) = 1 4, τ = 5. Then d ij = 1 c ij(t) L j b ij(t)l j ) i, j = 1, 2, i (t)(a ( ) 1/2 1/18 D = (d ij ) 2 2 =. 2 1/2 Hence, ρ(d) = 5/6 < 1. It follows from the theory of M-matrix in [9] that there exist constants η > 0 and ξ i > 0, i = 1, 2, such that for all t > 0, there holds c i (t)ξ i a ij(t) L j ξ j b ij(t) L j ξ j < η < 0, i = 1, 2.

7 EJDE-2007/46 CONVERGENCE BEHAVIOUR OF SOLUTIONS 7 Then, we can choose constants η > 0 and 0 < λ < 1 such that [c i (t) λ]ξ i a ij(t) L j ξ j b ij(t) e λτ L j ξ j < η < 0, i = 1, 2, t > 0, which implies that systems (3.1) and (3.2) satisfy (H1) (H3). Hence, from Lemma 1.1 and Theorem 2.1, system (3.2) has exactly one 2π-periodic solution. Moreover, all solutions of (3.1) converge to the periodic solution of (3.2). Remark 3.1. Since CNNs (3.1) is a delayed neural networks without periodic coefficients, the results in [3, 5, 4, 6, 7, 8, 9] and the references therein can not be applied to prove that all solutions converge to a periodic function. This implies that the results of this paper are essentially new nd they complement previously known results. References [1] A. Berman and R. J. Plemmons; Nonnegative Matrices in the Mathematical Science, Academic Press, New York, [2] L. O. Chua, T. Roska; Cellular neural networks with nonlinear and delay-type template elements, in: Proc IEEE Int. Workshop on Cellular Neural Networks and Their Applications, 1990, pp [3] J. Cao; New results concerning exponential stability and periodic solutions of delayed cellular neural networks with delays, Physics Letters A 307 (2003), [4] H. Huang, J. Cao and J. Wang; Global exponential stability and periodic solutions of recurrent cellular neural networks with delays, Physics Letters A 298 (5-6) (2002), [5] Q. Dong, K. Matsui and X. Huang; Existence and stability of periodic solutions for Hopfield neural network equations with periodic input, Nonlinear Analysis 49 (2002), [6] Z. Liu and L. Liao; Existence and global exponential stability of periodic solutions of cellular neural networks with time-vary delays, Journal of Mathematical Analysis And Applications, 290 (2) (2004), [7] B. Liu and L. Huang; Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays, Physics Letters A, 349 (2006), [8] W. Lu and T. Chen; On periodic Dynamical systems, Chinese Annals of Mathematics, B(25) (2004) [9] K. Yuan, J. Cao and J. Deng; Exponential stability and periodic solutions of fuzzy cellular neural networks with time-varying delays, Neurocomputing, 69 (2006), Bing Xiao Department of Mathematics, Hunan University of Arts and Science, Changde, Hunan , China address: bingxiao209@yahoo.com.cn Hong Zhang Department of Mathematics, Hunan University of Arts and Science, Changde, Hunan , China address: hongzhang320@yahoo.com.cn